For your second question...
In general, the graph of $y=f(2x)$ looks lượt thích the graph of $y=f(x)$, except that it is compressed horizontally around the $y$-axis by a factor of $2$ (the function "goes through" the values of $x$ twice as fast; so sánh $f$ takes all values it used to tát take between, say, $0$ and $2$, but does it between $0$ and $1$; so sánh what used to tát "take" all the way to tát $x=2$ to tát graph now needs to tát be graphed between $0$ and $1$, hence "twice as fast").
Likewise, the graph of $y=f(3x)$ is lượt thích the graph of $y=f(x)$, compressed by a factor of $3$; $y=f(4x)$ is lượt thích the graph of $y=f(x)$ compressed by a factor of $4$, etc. a
That means that the graph of $y=sin(2x)$ will look just lượt thích the graph of $y=\sin(x)$, but compressed by a factor of $2$. Since $y=\sin(x)$ takes $2\pi$ to tát bởi a full cycle, then $y=\sin(2x)$ will take half as long (will bởi the cycle "twice as fast"). So the period is $\pi$ instead of $2\pi$.
Similarly, the graph of $y=\cos(5x)$ is lượt thích the graph of $y=\cos(x)$, but compresssed by a factor of $5$; it finishes a cycle five times as fast as the regular cosine does; so sánh it does a cycle every $\frac{2\pi}{5}$.
If you have one function repeating every $\pi$, and one function repeating every $\frac{2\pi}{5}$, then they will "sync back up" at $2\pi$, because that is the smallest integer multiple of both $\pi$ and $\frac{2\pi}{5}$, as you surmise.
Alternatively: if $y=g(x)$ is periodic with period $P$, that means that $g(x+P)=g(x)$ for all $x$. What is the period of $y=g(2x)$? Well, if you instead of $x$ you plug in $x+\frac{P}{2}$, you get: $$y = g\left( 2\left(x+\frac{P}{2}\right)\right) = g(2x + P) = g(2x)$$ so $\frac{P}{2}$ is a period for $y=g(2x)$. Generally, if $y=g(x)$ has period $P$, then $y=g(kx)$ has period $\frac{P}{k}$ (can you see why?).